1 Separate variables and solve the differential equation a d

1. Separate variables and solve the differential equation:

a) dy/dx = y/x lnx

Solution

ans)dy/dx = ln(x) / (xy)

Multiply both sides by xy:
xy dy/dx = ln(x)

Separate variables on left-hand side:
x/dx * y dy = ln(x)

Multiply both sides by dx/x:
y dy = [ln(x) / x] dx

Integrate both sides:
y dy = [ln(x) / x] dx

On right-hand side:
Let u = ln(x)
du = dx / x

Make the substitution:
y dy = u du

Complete the integration on both sides:
y²/2 + C = u²/2 + C

Reverse substitute and combine arbitrary constants on both sides:
y²/2 = [ln(x)]²/2 + C

Multiply both sides by 2:
y² = [ln(x)]² + C

Plug in the initial condition to solve for C:
2² = [ln(1)]² + C
4 = 0 + C
C = 4

Solution with value for C:
y^2 = [ln(x)]² + 4

Take the square root of both side for solutions:
y = ± sqrt[ [ln(x)]² + 4 ]